Consider an economy consisting of two individuals. The utility of person i is

U_i = s - (1/2) (c_i)²

Where s is the quantity of public good, and c_i is person c_i's contribution to the provision of the public good (measured in private goods). In this economy, one unit of private goods can always be transformed into one unit of public goods, so 's' satisfies

c₁ + c₂ = s

a) What is each persons's Marginal Rate of Substitution (MRS) [Hint: MRS_i = - [MU^s/MU^c_i]?

b) What is the Samuelson Condition for this economy?

c) An efficient allocation is a triplet (s, c₁, c₂) which satisfies the resource constraint and Samuelson condition. Can you find such an allocation?